Every student has mathematical strengths beyond knowing basic facts, solving problems quickly, or showing work clearly. In this article, the author presents Smiles as an "on-ramp" task that supports students working together by unveiling and leveraging mathematical strengths. Nielsen describes on-ramp mathematics tasks as scaffolds that…

Secondary school students in Singapore are expected to find an expression for the general or "nth" term of an arithmetic progression (AP) without using the AP formula T[subscript n] = a + (n-1)d, where "a" is the first term, "n" is the number of terms and "d" is the common difference between successive…

Most future teachers are familiar with number patterns that represent an arithmetic sequence, and most are able to determine the general representation of the "n"th number in the pattern. However, when they are given a visual representation instead of the numbers in the pattern, it is not always easy for them to make the connection between the…

This note presents demonstrations of mathematics that emerges when problems are posed with respect to a combined 12 x 12 multiplication table showing multiplier and multiplicand. Through processes such as recognizing and extending patterns, specializing and generalizing particular functional relationships between the diagonal and row sequences are…

Magic squares have been of interest as a source of recreation for over 4,500 years. A magic square consists of a square array of n[squared] positive and distinct integers arranged so that the sum of any column, row, or main diagonal is the same. In particular, an array of consecutive integers from 1 to n[squared] forming an nxn magic square is…

Pattern recognition is a critical component of success in mathematics. Students at all levels should be provided with opportunities to investigate and uncover patterns throughout their mathematical careers. Further, they should be allowed to explore situations in which pattern recognition plays a vital role in the construction of important…